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Beta laws: arcsine, uniform, semicircle

The Beta law on \( {[0,1]} \) with parameters \( {a>0} \) and \( {b>0} \) has density

\[ x\mapsto \frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)} x^{a-1}(1-x)^{b-1}\mathbf{1}_{[0,1]}(x). \]

This family of laws allows to interpolate between the arcsine law \( {a=b=1/2} \) and the semicircle law \( {a=b=3/2} \), passing thru the uniform law \( {a=b=1} \). It is sometimes more convenient to work on the interval \( {[-1,1]} \) instead of \( {[0,1]} \). The Jacobi polynomials are orthogonal for this \( {(a,b)} \)-model. We recover the Chebyshev polynomials of the first and second kind in the arcsine and semicircle cases respectively.

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